Question

Solve que no. 16 please

Answer 1

let n be a positive integer

then,n can be odd or even .

i) if n is odd,it is not divisible by 2.
we can write n=1*n=2*n

=product of a non-negative power of 3 and an odd no.

ii)if n is even ,it is divisible by 2
then m=n/2 is an integer

if m is odd ,it cannot be divisible by 2

because m=n/2,n=2m=2¹*m

=product of a non negative power of 2 and an odd number

if m is even,it is divisible by 2.

then p=m/2 is an integer

If p is odd ,it cannot be divisible by 2.

because p=m/2and m=n/2,we get p=n/4,or n=4p=2²*p

=product of a non- negative power of 2 and an odd no.

if p is even it is further divisible by 2 ,and the above steps can be repeated until we arrive at an integer which is no longer divisible by 2,i.e.,it is odd

........................
hence proved that every positive integer is uniquely representable as the product of a non-negative power of 2 and an odd no.